An evaluation of the data space dimension in phase retrieval: results in Fresnel zone

In this paper, we address the problem of computing the dimension of data space in phase retrieval problem. Starting from the quadratic formulation of the phase retrieval, the analysis is performed in two steps. First, we exploit the lifting technique to obtain a linear representation of the data. Later, we evaluate the dimension of data space by computing analytically the number of relevant singular values of the linear operator that represents the data. The study is done with reference to a 2D scalar geometry consisting of an electric current strip whose square amplitude of the electric radiated field is observed on a twodimensional extended domain in Fresnel zone

On the Sampling of the Fresnel Field Intensity over a Full Angular Sector

In this article, the question of how to sample the square amplitude of the radiated field in the framework of phaseless antenna diagnostics is addressed. In particular, the goal of the article is to find a discretization scheme that exploits a non-redundant number of samples and returns a discrete model whose mathematical properties are similar to those of the continuous one. To this end, at first, the lifting technique is used to obtain a linear representation of the square amplitude of the radiated field. Later, a discretization scheme based on the Shannon sampling theorem is exploited to discretize the continuous model. More in detail, the kernel of the related eigenvalue problem is first recast as the Fourier transform of a window function, and after, it is evaluated. Finally, the sampling theory approach is applied to obtain a discrete model whose singular values approximate all the relevant singular values of the continuous linear model. The study refers to a strip source whose square magnitude of the radiated field is observed in the Fresnel zone over a 2D observation domain

The Dimension of Phaseless Near-Field Data by Asymptotic Investigation of the Lifting Operator

In this paper, the question of evaluating the dimension of data space in an inverse source problem from near-field phaseless data is addressed. The study is developed for a 2D scalar geometry made up by a magnetic current strip whose square magnitude of the radiated field is observed in near non-reactive zone on multiple lines parallel to the source. With the aim of estimating the dimension of data space, at first, the lifting technique is exploited to recast the quadratic model as a linear one. After, the singular values decomposition of such linear operator is introduced. Finally, the dimension of data space is evaluated by quantifying the number of “relevant” singular values. In the last part of the article, some numerical simulations that corroborate the analytical estimation of data space dimension are shown

A sampling strategy of the radiation operator in near-zone based on an asymptotic kernel

In this paper, we address the problem of discretizing the singular system of the radiation operator concerning the case of a magnetic strip current whose radiated field is observed in near-zone on a bounded line parallel to the source. This question has been already addressed in previous articles with the limitation that the extension of the observation domain does not overcome the source size. In this article, we remove such limitation, hence, we provide a discrete model that well approximates the singular values of the radiation operator in the case where the observation domain is larger than the source

Allele-Specific HLA Loss and Immune Escape in Lung Cancer Evolution

Immune evasion is a hallmark of cancer. Losing the ability to present neoantigens through human leukocyte antigen (HLA) loss may facilitate immune evasion. However, the polymorphic nature of the locus has precluded accurate HLA copy-number analysis. Here, we present loss of heterozygosity in human leukocyte antigen (LOHHLA), a computational tool to determine HLA allele-specific copy number from sequencing data. Using LOHHLA, we find that HLA LOH occurs in 40% of non-small-cell lung cancers (NSCLCs) and is associated with a high subclonal neoantigen burden, APOBEC-mediated mutagenesis, upregulation of cytolytic activity, and PD-L1 positivity. The focal nature of HLA LOH alterations, their subclonal frequencies, enrichment in metastatic sites, and occurrence as parallel events suggests that HLA LOH is an immune escape mechanism that is subject to strong microenvironmental selection pressures later in tumor evolution. Characterizing HLA LOH with LOHHLA refines neoantigen prediction and may have implications for our understanding of resistance mechanisms and immunotherapeutic approaches targeting neoantigens. Video Abstract [Figure presented] Development of the bioinformatics tool LOHHLA allows precise measurement of allele-specific HLA copy number, improves the accuracy in neoantigen prediction, and uncovers insights into how immune escape contributes to tumor evolution in non-small-cell lung cancer

Performance of Phase Retrieval via Phaselift and Quadratic Inversion in Circular Scanning Case

The reconstruction of the field radiated by a source from square amplitude-only data falls into the realm of phase retrieval. In this paper, we tackle the phase retrieval with two different approaches. The first one is based on a convex optimization problem called PhaseLift. The latter exploits the lifting technique to recast phase retrieval as a linear problem with an increased number of unknowns and then, because the linear problem is highly undetermined, it adds further constraints (based on the mathematical properties of the solution) to estimate it. The second approach formulates phase retrieval as a least squares problem and, therefore, it requires to tackle the minimization of a quartic functional which will be carried out by applying a gradient descent method. In the second part of this paper, in order to corroborate the effectiveness of both approaches, we present the numerical results. Afterward, we provide a comparison between the two methods and finally, we emphasize how the ratio between the number of independent data and the number of unknowns impacts on the performance in both approaches

An SVD approach for estimating the dimension of phaseless data on multiple arcs in Fresnel zone

In this article, we tackle the question of evaluating the dimension of the data space in the phase retrieval problem. With the aim to achieve this task, we first exploit the lifting technique to recast the quadratic model as a linear one. After that, we evaluate analytically the singular values of the lifting operator, and we quantify the dimension of the data space by counting the number of “significant” singular values. In the last part of the article, we show some numerical results in order to corroborate our analytical prediction on the singular values’ behavior of the lifting operator and on the dimension of the data space. The analysis is performed for a 2D scalar geometry consisting of an electric current strip whose square magnitude of the radiated field is observed on multiple arcs of circumference in Fresnel zone

On data increasing in phase retrieval via quadratic inversion: flattening manifold and local minima

In this paper, we address the question of traps in phase retrieval via quadratic inversion. First, we clarify the genesis of local minima from a geometrical perspective by showing how the shape of the manifold of data and the position of the data point are related to the presence of local minima in the objective functional. Later, we recall a mathematical condition for the lack of traps. Afterwards, we illustrate how the ratio between the dimension of data space M and the number of real unknowns Nr impacts on the manifold of data and consequently on the presence of local minima. In particular, we show that if the dimension of data space is high enough, the manifold of data is sufficiently large and no traps appear in the functional. In the last part of the paper, with reference to a particular unknown sequence, we establish the minimum value of the ratio M / Nr such that no traps should appear in the functional. In this condition phase retrieval via quadratic inversion exhibits a global convergence behavior, consequently, the actual solution of the problem can be reached regardless of the initial guess which can be chosen completely at random

Asymptotic Study of the Radiation Operator for the Strip Current in Near Zone

In this paper, we address the problem of how to efficiently sample the radiated field in the framework of near-field measurement techniques. In particular, the aim of the article is to find a sampling strategy for which the discretized model exhibits the same singular values of the continuous problem. The study is done with reference to a strip current whose radiated electric field is observed in the near zone over a bounded line parallel to the source. Differently from far-zone configurations, the kernel of the related eigenvalue problem is not of convolution type, and not band-limited. Hence, the sampling-theory approach cannot be directly applied to establish how to efficiently collect the data. In order to surmount this drawback, we first use an asymptotic approach to explicit the kernel of the eigenvalue problem. After, by exploiting a warping technique, we recast the original eigenvalue problem in a new one. The latter, if the observation domain is not too large, involves a convolution operator with a band-limited kernel. Hence, in this case the sampling-theory approach can be applied, and the optimal locations of the sampling points can be found. Differently, if the observation domain is very extended, the kernel of the new eigenvalue problem is still not convolution. In this last case, in order to establish how to discretize the continuous model, we perform a numerical analysis

NDF of the Near-Zone Field on a Line Perpendicular to the Source

In this paper, the problem of computing the number of degrees of freedom (NDF) of the field radiated by a strip current along all the possible lines orthogonal to the source is addressed. As well known, the NDF is equal to the number of singular values of the radiation operator that are before a critical index at which they abrupt decay. Unfortunately, in the considered case, the solution of the associate eigenvalue problem is not known in closed-form, and this prevents us from directly evaluating the singular values of the radiation operator. To overcome this drawback, a weighted adjoint operator is exploited. The latter allows obtaining an eigenvalue problem whose solution is known in closed-form but, at the same time, it modifies the behavior of the singular values. However, since the change affects only the dynamics of the singular values but not the critical index at which they abrupt decay, the NDF of the radiated field can be analytically estimated by resorting to the weighted adjoint operator